Problem: $ 0.\overline{73} \div 0.\overline{4} = {?} $
Explanation: First convert the repeating decimals to fractions. $\begin{align*} 100x &= 73.7373...\\ x &= 0.7373...\end{align*} $ $\begin{align*} 99x &= 73 \\ x &= \dfrac{73}{99}\end{align*} $ $\begin{align*} 10y &= 4.4444...\\ y &= 0.4444...\end{align*} $ $\begin{align*} 9y &= 4 \\ y &= \dfrac{4}{9}\end{align*} $ So, the problem becomes: $ \dfrac{73}{99} \div \dfrac{4}{9} = {?} $ Dividing by a fraction is the same as multiply by the reciprocal of that fraction. $ \dfrac{73}{99} \times \dfrac{9}{4} = {?} $ $ \phantom{\dfrac{73}{99} \times \dfrac{4}{9}} = \dfrac{73 \times 9}{99 \times 4} $ $ \phantom{\dfrac{73}{99} \times \dfrac{4}{9}} = \dfrac{73 \times \cancel{9}} {\cancel{99}11 \times 4} $ $ \phantom{\dfrac{73}{99} \times \dfrac{4}{9}} = \dfrac{73}{44} $